Weighted Random Popular Matchings

For a set A of n applicants and a set I of m items, we consider a problem of computing a matching of applicants to items, i.e., a function M mapping A to I; here we assume that each applicant $x \in A$ provides a preference list on items in I. We say that an applicant $x \in A$ prefers an item p than an item q if p is located at a higher position than q in its preference list, and we say that x prefers a matching M over a matching M' if x prefers M(x) over M'(x). For a given matching problem A, I, and preference lists, we say that M is more popular than M' if the number of applicants preferring M over M' is larger than that of applicants preferring M' over M, and M is called a popular matching if there is no other matching that is more popular than M. Here we consider the situation that A is partitioned into $A_{1},A_{2},...,A_{k}$, and that each $A_{i}$ is assigned a weight $w_{i}>0$ such that w_{1}>w_{2}>...>w_{k}>0$. For such a matching problem, we say that M is more popular than M' if the total weight of applicants preferring M over M' is larger than that of applicants preferring M' over M, and we call M an k-weighted popular matching if there is no other matching that is more popular than M. In this paper, we analyze the 2-weighted matching problem, and we show that (lower bound) if$m/n^{4/3}=o(1)$, then a random instance of the 2-weighted matching problem with$w_{1} \geq 2w_{2}$has a 2-weighted popular matching with probability o(1); and (upper bound) if$n^{4/3}/m = o(1)$, then a random instance of the 2-weighted matching problem with$w_{1} \geq 2w_{2}$ has a 2-weighted popular matching with probability 1-o(1).Comment: 13 pages, 2 figure

Tectonic Synthesis: A Plate Reconstruction Model of the NW Pacific Region Since 100 Ma

Based on the results of interdisciplinary study from Chapters 1–4, a plate tectonic model of the northwestern Pacific region since 100 Ma is presented in this chapter. The evolution of the Pacific margin is viewed as a longstanding history of migration/amalgamation of allochthonous blocks onto the subduction zone. Such a process inevitably provoked diverse tectonic events, spatiotemporal positions of which have been discussed in this book. In order to reconcile paradoxical discrepancies in the docking process of arc fragments, the authors introduce a marginal sea plate with a spreading center that was alive in the Cretaceous. Oblique subduction of the ridge caused specific migratory igneous activity along the rim of the overriding plates, together with flips of shearing direction. Arc-trench systems on the eastern and western sides of the marginal sea plate developed following different timelines and were eventually mixed up during the plate’s closure that prompted formation of a coincident Oligocene clinounconformity widespread on the Eurasian margin. Since the demise of the hypothetical plate, the tectonic regime of the northwestern Pacific margin has been controlled by the growth, namely, the rotational history and modes of convergence of the Philippine Sea Plate

Retail Planning Policy and Recent Regional Trends of Retail Activities in Auckland City, New Zealand

本稿はニュージーランドオークランド市の小売商業の計画政策と小売商業の地域的動向についてみたものである。オークランド市の小売商業の計画政策は，ゾーニング制度を用いて小売商業の開発と立地をコントロールすることで，小売商業地の階層的地域体系と既存の小売商業地区を存続・維持するとともに，急速な人口増加に対しては，新設センターの開発と既存のセンターの周辺に小売商業地区を拡張する用地を担保することで地域的需給ギャップの調整を図っている。また，近年の地域小売商業の動向としては，大規模なセンターや新設センターが成長・発展するとともに，既存の小売商業地区のなかには，マイノリティ人口の増加に対応して，エスニックマーケットへと変容してきたところもみられてきている。This paper aims to examine the retail planning policies and the recent regional trends of the retail activities in Auckland, New Zealand. Its retail planning policy is expressed in "Auckland Plan" which was approved in 2012. The main purpose of this plan is to control the retail development by land use zoning and to maintain the regional hierarchy of the existing retail centres in order to keep their vitality. The regional gap between supply and demand due to the rapid population growth is adjusted by the developing the emergent centres and the reserving additional land for the future development of the existing centres. From considering the recent retail activities, it became clear that the larger centres and the emergent centres have been growing faster than the others and some existing centres in the area with rapid ethnic population growth have been transforming to the ethnic market centres.伊東理教授退職記念号特集 : 都市空間の地理